JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 90, NO. C2, PAGES 3332-3342, MARCH 20, 1985

Physical Properties of Seawater' A New Salinity Scale and Equation of State for Seawater

N. P. FOFONOFF

Woods Hole Oceanographic Institution, Massachusetts

Developments leading to the new practical salinity scale and equation of state for seawater, introduced by the Joint Panel on Oceanographic Tables and Standards (JPOTS), are reviewed. The laboratory measurements used to construct the new empirical formulas for converting conductivity ratios to salinity and for estimating seawater density from salinity, temperature, and pressure were made on samples of standard seawater to minimize the effects of compositional variations. Measured densities were fitted with a standard deviation of 0.0036 kg/m 3 at atmospheric pressure and 0.009 kg/m 3 over the oceanic pressure range. Natural seawater densities can differ from computed density by as much as 0.05 kg/m 3 because of variations in the composition of dissolved salts that are not reflected equivalently in measured conductivity ratios. Differences from the previous Knudsen-Ekman equation of state are also of com- parable magnitude. Values of specific volume anomaly for geopotential anomaly computations for esti- mation of geostrophic currents and transport are not significantly modified. Algorithms adopted by the JPOTS for computing adiabatic lapse rate and potential temperature are compared with formulas derived from the new equation of state. Potential temperatures agree within 2 x 10-3øC over the oceanic range of temperature and pressure at salinity of 35.

HISTORICAL BACKGROUND

The completion in 1981 of a series of recommendations and reports by the Joint Panel on Oceanographic Tables and Standards (JPOTS) on the definition of a practical salinity scale (PSS 78) and an equation of state for seawater (EOS 80) marked a milestone in the establishment of precise and repro- ducible standards for salinity, density, and related variables for ocean waters.

Because JPOTS introduced a new salinity scale based on constant composition of salts and expressed in terms of an electrical conductivity ratio, consistent algorithms had to be developed for other derived or dependent functions. Formulas for a few of the more commonly used physical properties have been tabulated by Fofonoff and Millard [1984]. These are dis- cussed in the present article together with a review of some of the considerations leading to the new definitions and algo- rithms. The notation and definitions are similar to those used

by Fofonoff [1962]. The units and symbols follow recommen- dations of the IAPSO (International Association for Physical Sciences of the Oceans) working group on symbols, units, and nomenclature [IAPSO, 1979].

The evolution of the practical salinity scale and equation of state was initiated in part by Eckart's [1958] concern about the accuracy of knowledge of fundamental properties such as density and thermal expansion of seawater. His examination of the available data led him to conclude that thermal ex-

pansion coefficients of seawater and even pure water were not known with acceptable accuracy. With the cooperation and support of the U.S. National Academy of Sciences Committee on Oceanography and the Office of Naval Research, he con- vened a conference to examine the state of knowledge of the chemical and physical properties of seawater with the aim of stimulating and focusing further research in this area. Among the recommendations included in the conference report [NAS- NRC, 1958] were steps to label and certify Copenhagen Stan- dard Seawater for electrical conductivity and density as well

Copyright 1985 by the American Geophysical Union.

Paper number 4C1335. 0148-0227/85/004C- 1335505.00

as chlorinity. These recommendations were made in recogni- tion of the increasing use of conductivity bridges (sali- nometers) for salinity determination and increasing concern about the adequacy of a chlorinity standard to serve as a conductivity standard.

At the same conference, Reid [1958] presented a compre- hensive analysis showing the dependence of geostrophic and dynamic computations on errors in the equation of state. He computed the sensitivity to thermal expansion and the saline contraction coefficients to demonstrate that dynamic compu- tations were not limited by inaccuracies of the coefficients. For other calculations, such as stability or Brunt-Vfiisfilfi fre- quency, accurate coefficients are a necessity.

A study of the relationships among chlorinity, conductivity, and density was undertaken at the British National Institute of Oceanography in 1960 on several hundred samples of sea- water collected from widely distributed locations over the world oceans. Cox et al. [1962] reported a markedly higher correlation between density and conductivity than between density and chlorinity. These findings tended to confirm the misgivings about the methods for standardizing salinity measurements and led to the formation of a Joint Panel on

the Equation of State of Seawater, sponsored by ICES (Inter- national Council for the Exploration of the Sea), IAPSO, SCOR (Scientific Committee on Oceanic Research) and UNESCO (renamed the Joint Panel on Oceanographic Tables and Standards (JPOTS) in 1964). The panel's task was to review the relationship of the equation of state to chlorinity, salinity, electrical conductivity, and refractive index to see if redefinition of any of these properties was necessary. The panel members concluded, after reviewing the data presented by Cox [1962; reprinted in UNESCO, 1976a, Append. 1], that salinity should not be defined in terms of chlorinity because of the excessive variability of its relationship to density. Al- though some of the variability later proved to be spurious and a consequence of the laboratory technique used to determine chlorinity [Cox et al., 1967], the scatter consistently exceeded the precision of measurement, supporting the possibility that density could be estimated more precisely from conductivity than from chlorinity.

Because of the better correlation between conductivity and 3332

FOFONOFF: PRACTICE OF OCEANOGRAPHY 3333

density, the initial panel recommendation was to establish an equation defining salinity in terms of density. Different tech- niques of measurement could then be implemented by con- struction of formulas relating density to a measured variable such as electrical conductivity, refractive index, or density itselfi Because density can be measured, whereas salinity cannot, the link to salinity would be through the defining density equation. Salinity must be proportional to chlorinity to retain its conservative property and to have a proportion- ality factor of 1.80655 to match historical data. Subsequent determinations of salinity of standard seawater (SSW) would be made from measurements of density or conductivity but not chlorinity, which would be reported as an independent variable.

The recommendations were made with the assumption that absolute density and conductivity of SSW could be measured with the required precision. However, technical problems en- countered in constructing equipment to achieve the precision necessary for acceptable standards delayed implementation of the recommendations. Relative measurements were considered

and dismissed because the references could not be defined with

sufficient precision. Distilled water, for example, differed in density, depending on its origin and method of preparation. The uncertainties in density were believed to be as large as 30 parts per million [UNESCO, 1976a, Append. 2]. Similarly, conductivity references, such as potassium chloride solutions, required temperature regulation that was not readily achiev- able [Cox, 1962]. As a result, only part of the recommen- dations could be implemented. Salinity was defined in terms of the electrical conductivity ratio R•5 at 15øC to seawater having a salinity of 35 [UNESCO, 1966, 1976b]. The reference seawater salinity was not defined separately in terms of con- ductivity ratio or density.

Tables to convert conductivity ratio to salinity, based on a regression formula developed by Cox [1962], were published by NIO-UNESCO [1966] and widely distributed to oceanog- raphers with the understanding that SSW would be certified for conductivity as soon as the necessary equipment was com- pleted. The new scale was officially adopted after Cox's death in 1967 and was formally announced in 1969 [Wooster et al., 1969].

Objections to the 1969 scale were raised. Park [1964] had already noted that conductivity of different batches of SSW measured relative to a single reference batch varied in excess of the precision of conductivity measurements and had called for certification of conductivity as well as chlorinity for SSW.

Other objections surfaced. The determinations of R•5 [Cox et al., 1967] were made on natural seawater samples of vari- able composition. Some midrange salinities were obtained by mixing Baltic and Red Sea samples. As the salinity-chlorinity conversions assumed fixed composition, the salinities were not accurately conservative. As salt ratios of natural seawaters are time dependent, especially at low salinities, the scale is not accurately reproducible. Samples from the same geographical locations could not be relied upon to yield the same relation- ship between conductivity and chlorinity. The increasing use of in situ instruments measuring conductivity forced the use of conversion formulas that were obtained by using standard seawater diluted with distilled water [Brown and Allentoil, 1966]. These formulas were not consistent with the JPOTS versions, which did not cover the temperature range below 10øC. Some of the problems associated with the 1969 scale were addressed in the sixth JPOTS report [UNESCO, 1974] with recommendations to measure the dependence of conduc-

tivity on temperature and pressure. Also, to have an indepen- dent check on the absolute conductivity apparatus, Poisson was encouraged to make some measurements of electrical con- ductivity of standard seawater samples relative to potassium chloride solutions to compare with the absolute measurements planned at National Institute of Oceanography (U.K.) by Culkin. Poisson was able to report at a subsequent JPOTS meeting [UNESCO, 1976a] that deviations of conductivity equivalent to 0.006 salinity were found for different batches of SSW. The panel viewed these results with some consternation and called for confirmatory studies. Several laboratories were asked subsequently to measure conductivity of 26 batches of SSW relative to P64. These studies [Millero et al., 1977] con- firmed Poisson's findings. Some of the batches were high in conductivity by amounts equivalent to 0.004 to 0.007 salinity [UNESCO, 1978]. Poisson reported at the same meeting of JPOTS that potassium chloride solutions could be made with high and reproducible accuracy. Furthermore, the temper- ature control required for thermostatted baths was well within the capability of commercially available equipment. Thus the reservations about accepting potassium chloride as a conduc- tivity standard were removed. The panel accepted a rec- ommendation to use potassium chloride solution as a stan- dard and outlined a procedure to establish the salinity scale. The determinations were to be made on a single batch of SSW to ensure constancy of composition. The scale was to be relat- ed to the 1969 and previous scales at 35 by the chlorinity factor 1.80655 for that particular batch. For subsequent batches the salinity was to be determined from the conduc- tivity ratio to the reference KC1 standard and not from chlori- nity. Hence the chlorinity factor is not a constant but would vary slightly from batch to batch.

The steps that followed to define the practical salinity scale are thoroughly documented in the eighth, ninth, and tenth reports of JPOTS [UNESCO, 1978, 1979, 1981c] and col- lected background papers [UNESCO, 1981b].

In addition to the definition of the salinity scale, the panel encouraged the development of algorithms for converting in situ measurements of conductivity ratio to salinity. These al- gorithms were developed to be consistent with the new practi- cal salinity.

THE PRACTICAL SALINITY SCALE (PSS 78)

Prior to the introduction of the 1969 salinity scale, salinity $ was expressed in terms of chlorinity C1, defined as the chloride-equivalent mass ratio of halides to the mass of seawa- ter, by a linear least squares regression formula fitted to mass ratios of salt residues of nine seawater samples evaporated to dryness [Knudsen et al., 1902]. The relationship $ = 0.030 + 1.8050 C1 served oceanographers for over six decades. The formula for $, commonly referred to as the Knudsen salinity, incorporated changing composition of seawater with dilution and consequently was not conservative with respect to addi- tion or removal of water. As long as salinities were computed from chlorinities, the Knudsen scale was acceptable. The in- creasing use, during the 1950's, of high-precision electrical conductivity bridges for salinity determinations created the need for new standards for salinity measurements.

The 1969 scale, proportional to chlorinity to satisfy mass conservation, was chosen to coincide with the Knudsen scale at salinity of 35, yielding the factor 1.80655 = 1.8050135/(35-0.030)]. Differences between the two scales are less than 0.003 over the salinity range 32-38 and increase to 0.03 at low salinity. As noted earlier, the 1969 scale was

3334 FOFONOFF: PRACTICE OF OCEANOGRAPHY

TABLE 1. Concentration of Potassium Chloride Solutions Having the Same Conductivity as Seawater (Batch P79) at Salinity S = 35 (PSS 78)

Standard Deviation Standard Deviation

Concentration of Residuals, Number of of Mean, Reference g/kg g/kg Measurements g/kg

Dauphinee et al. [1980a] 32.4356 3.6 x 10 -'• 21 0.8 x 10 -'• Poisson [1980a] 32.4358 4.7 x 10 -'• 14 1.3 x 10 -'• Culkin and Smith [1980] 32.4352 (3.4 x 10 -'•) 17 (0.8 x 10 -'•) S-- 35 (PSS 78) 32.4356 0 (by definition)

Values in parentheses are calculated from published table.

deficient in not establishing a salinity reference point indepen- dent of chlorinity and inconsistent in using seawater samples of variable composition to relate salinity to electrical conduc- tivity.

The fundamental step in constructing the practical salinity scale PSS 78 consisted of defining a single reference point (S = 35) on the scale as having the same electrical conductivity as a reference potassium chloride (KC1) solution at 15øC and atmospheric pressure. The transition from the previous scale was made by selecting a single batch (P79) of SSW and equat- ing the new scale to the old through the chlorinity relationship S = 1.80655 C1 for that particular batch. As the salinity on the practical scale is defined to be conservative with respect to addition and removal of water, the entire salinity range is accessible through precise weight dilution or evaporation without additional definitions. However, the practical scale is defined in terms of conductivity ratio and not by its conserva- tive properties. Thus the second part of the definition was constructed by measuring the conductivity ratio to the KC1 standard or an equivalent secondary standard over the entire range of salinities (1 to 42) of samples prepared by evapora- tion or dilution of batch P79 SSW and computing an empiri- cal formula $ -- S(R•5), where R•5 is the conductivity ratio at 15øC and atmospheric pressure to the KC1 standard. Thus salinities on the PSS 78 scale are defined by conductivity ratios alone. Salinities determined by any other method would not necessarily coincide with the PSS 78 scale and would have to be identified separately. Lewis and Perkin [1981] discuss differences between PSS 78 and previous scales and provide algorithms and tables to convert existing data to the new scale.

The KC1 reference solution for S- 35 was established by three independent sets of measurements by Dauphinee et al. [1980a], Poisson [1980a], and Culkin and Smith [1980]. The values obtained and the precision of the determinations are summarized in Table 1. Both standard deviations of residuals

and of the means are computed. The means given by Daup- hinee et al. [1980a] and Poisson [1980a] are not significantly different from each other but are significantly different (3 to 4 standard deviations) from the mean value obtained by Culkin and Smith. Thus, although all three determinations are suf- ficiently accurate to define the reference an order of magnitude more precisely than required for routine CTD instruments, statistically significant differences did appear for the absolute measurements of Culkin and Smith.

The data sets used by Perkin and Lewis [1980] to express the dependence of conductivity ratio on temperature and pres- sure are listed in Table 2. All of the measurements were made

by using PSS 78 and batches P73, P75, and P79 of SSW. Differences in composition between the batches would have negligible effects on the temperature and pressure coefficients of conductivity ratios, as neither are strong functions of salini- ty. Mantyla [1980] measured the conductivity ratios of 39 different batches of SSW relative to P79. Batches P73 and P75

yielded differences of -0.0011 and +0.0005, respectively, be- tween conductivity and chlorinity salinity. As the standard deviation of replicates was 0.0004, the batches were not dis- tinguishable from one another in composition.

The algorithm for converting conductivity ratio to salinity is constructed in terms of the conductivity ratio R, defined as

R = C(S, t, p)/C(35, 15, 0) (1)

TABLE 2. Precision of Empirical Formulas Relating Salinity and Conductivity Ratios Rt, rt, and R v

Standard Deviation

of Residuals, Number of Reference (Salinity Equivalent) Observations SSW Batch

Dauphinee et al. [1980a] Poisson [ 1980a] Perkin and Lewis [1980]

Dauphinee et al. [1980b] Poisson [1980b] Bradshaw and Schleicher [1980] Perkin and Lewis [1980]

Dauphinee et al. [1980b] Bradshaw and Schleicher [1980] Perkin and Lewis [1980]

Bradshaw and Schleicher [1980]

S = S(R15 ) 3.6 x 10 -'• 116 P79

4 x 10 -'• 38 P75 4.8 x 10 -'• combined fit

S = S(t,Rt) 189 P79

10 x 10 -'• 115 P75 8 x 10 -'• 101 P75

6.6 x 10 -'• combined fit

Pt-- Pt(t) 4.4 x 10 -'• 34 P79

3 x 10 -'• 36 P73, P75 3.2 x 10 -'• combined fit

R v = Rv(R,t,p) 13 x 10 -'• 224 P75

FOFONOFF: PRACTICE OF OCEANOGRAPHY 3335

where C(S, t, p) is electrical conductivity as a function of salin- ity S, temperature t, and pressure p. The ratio is factored into the functions

where

R = rt(t)Rt(S, t)R•,(R, t, p) (2)

r,(t) = C(35, t, 0)/C(35, 15, 0)

Rt(S, t)= C(S, t, 0)/C(35, t, 0)

Rt,(R, t, p)= C(S, t, p)/C(S, t, O)

JPOTS has not accepted or recommended a value for con- ductivity C(35, 15, 0) to avoid any confusion with conductivity ratio in the definition of practical salinity. For those appli- cations where absolute conductivity is required, a value of 4.29140 S/m (42.9140 mmho/cm) can be used to convert con- ductivity ratio to electrical conductivity. This value was ob- tained by Culkin and Smith [1980] using absolute conductivity equipment at the British Institute of Ocean Sciences.

Salinity is given by the function

S = an +- bn Rt n/2 (3) n: o 1 + kAt

with coefficients

ao = +0.0080

a• = -0.1692

a 2 -- + 25.3851

a 3 = + 14.0941

a 4 = -7.0261

as = +2.7081

At t = 15øC, (3) reduces to salinity scale [-Perkin and Lewis, 1980].

The factors r t and R, are given by

r t -- • Cn tn n=O

where

bo = +0.0005 k = +0.0162

b•= -0.0056 At=t--15

b 2 = -0.0066

b 3 :- -0.0375

b 4 = +0.0636

b 5 = -0.0144

the formula defining the practical

and

Co = +0.6766097 C 3

C• = +2.00564E-2 C•

C2 - + 1.104259E-4

with

= - 6.9698E-7

= + 1.0031E-9

Rp=l+ e•p + e2p 2 + e3p 3

1+

1 + d•t + d2 t2 + (d 3 + d4t)R

C

B+AR

(4)

e I = + 2.070E-5 d 1 -- +3.426E-2

e 2 ----6.370E-10 d2 = +4.464E-4

e3 = +3.989E-15 d3 = +4.215E-1

d,• = -3.107E-3

for temperature in degrees Celsius and pressure in decibars. Given a measurement of R, t, and p, salinity is computed by

solving (2) for R t' Rt = R/(rtRp) and evaluating S from (3). If

conductivity ratio R is required, given S, t, and p, the ratio Rt can be found by numerical inversion of (3), and R can be found by solving the quadratic equation

or

R = rtRtR•,= rtRt 1 + AR + B

R = {[(ArtR t -- B) 2 + 4rtRtA(B + C)] •/2 + [ArtR t -- B]}/2A

(6)

More detailed descriptions, including FORTRAN pro- grams, check values, limits of validity, and tables are given by Fofonoff and Millard [ 1984].

IDEAL OR ABSOLUTE SALINITY

Absolute or ideal salinity, defined simply as the mass frac- tion of salts in seawater, is important conceptually for formu- lating rigorous descriptions and precise assumptions about the dynamical and thermodynamical behavior of a seawater system. In its 1978 recommendations, JPOTS proposed use of the symbol S A to denote absolute salinity to distinguish it from practical salinity S. The practical salinity must be pro- portional to SA (or s used here) to represent conservation or salt mass in seawater.

The aggregate variable, salinity s, is useful because major ions are in nearly constant proportions to one another throughout the world oceans. This "law of constant propor- tions," first enunciated by Marcet [1819] and elaborated and documented more quantitatively by Forchhammer [1865] and Dirtmar [1884], appears to hold, with some exceptions, to within experimental error [Culkin, 1965]. However, the error of determining some of the major constituents separately is relatively large, so that a rigorous test to the same precision as conductivity measurements is difficult to carry out. Exceptions to the constancy of composition are found in coastal and enclosed regions, such as the Baltic Sea, diluted by land drain- age and surface waters in which biological processes can alter the composition significantly. Culkin and Cox [1966] noted that the concentration of calcium is the major variable com- ponent affecting density and conductivity between surface and deep waters of the open ocean. As real variations in compo- sition are present, the effects of salt content in seawater cannot be completely described by a single variable (salinity) within the framework of the hydrodynamic equations. The ideal sa- linity has limitations that require discussion. The first is that no convenient, inexpensive, and accurate techniques exist for determining total salt content in seawater. Estimates of salini- ty, obtained indirectly from measurements of chlorinity, elec- trical conductivity and, less commonly, specific gravity or re- fractive index, yield estimates of salinity on practical scales, such as PSS 78 discussed earlier, whose precise relationships to absolute salinity are unknown. The second is that an advective-diffusive conservation equation cannot be written for salinity alone because diffusion does not change the salt distribution toward constant composition or uniform salinity. The law of constant proportions is a consequence of dynam- ical processes of advection and turbulent mixing rather than a trend to thermodynamical equilibrium. This conclusion is de- veloped further by examining the conservation equations for a multicomponent thermodynamical system.

Suppose that a mass m i Of the ith solute constituent is pres- ent in a mass m of seawater consisting of mw water and m s =

3336 FOFONOFF: PRACTICE OF OCEANOGRAPHY

y', m i of solutes. The mass fraction of solutes or ideal salinity s is defined to be

s = ms/(m w + ms)= ms/m (7)

and the partial salinity (concentration) si = m•/m of the ith solute so that

s = Z s, (8)

If ;• is the ratio of the ith constituent to the total solutes so that si = 2is, the law of constant proportions implies that ;•i is independent of s.

A mass conservation equation must hold for each constit- uent mi. If Pi is the partial density of the ith constituent and Vi its velocity, the individual conservation equations take the form

-- q- V•oiV i : 0

C•pw •t

-- + VpwV w = 0

(9)

where the subscript w denotes the water or solvent compo- nent. The sum of the individual conservation equations leads to the usual continuity equation

provided

•p + vp v = 0 (10)

v = (Z + wVw)/

= • siVi + SwVw

are defined as the density p and velocity V of seawater. The concentrations si of solutes and Sw of water are related by

•si+Sw= 1 (12)

The individual equations can be rewritten relative to the mean velocity V as

c•pi • + VpiV = -Vp(Vi - V) = -VFi (13) at

where

F i = pi(Vi- V)- psi(V i - V) (14)

represents the flux or diffusion of the ith solute relative to the total mass flux. The driving force for diffusion of the ith com- ponent is assumed to be the gradient of its partial chemical potential Vbti. If the flux is denoted by J• = psiV•, the product Ji' V/-ti represents the rate of energy conversion for the ith component associated with the flux. The sum (y', Ji'Vl, ti -3- Jw' Vbtw) represents a rate of energy conversion D. As only the relative or diffusion fluxes contribute, the conversion can be interpreted as a dissipation rate of energy D or production of entropy D/T, where T is absolute temperature. Using the Gibbs-Duhem equation [Katchalsky and Curran, 1967] and neglecting thermal effects, the dissipation rate for fluxes F•, relative to the mass flux p¾, can be put in the form

D = -- 5', Ji'V].li - Jw'VPw

= - Z F,. V(#, - (iS)

For diffusive fluxes the entropy production on dissipation is a

positive definite function. The fluxes will be linear functions of the form

Fi = 5', v,kV(lak- 3tw) (16) k

The effects of diffusion are to diminish the gradients of bt i - #w at rates that can vary from solute to solute. Caldwell and Eide [1981] found variations of a factor of 2 or more in the diffu- sivities of major salts in seawater. Although diffusive fluxes must be further constrained and modified by requirement of balance of electrical charges and by variations of pressure and temperature, it is clear that diffusion does not act to preserve constancy of salt ratios but rather to equilibrate partial chemi- cal potentials.

Fofonoff[1962] estimated that the salinity gradient required to equalize the pressure dependence of chemical potential dif- ference bt = • ;•bq- btw = bts- btw is about 3 to 4 per 1000 decibars. As vertical gradients of this magnitude are not ob- served in the ocean except over short vertical scales, the diffu- sive fluxes do not appear to play a significant role compared with advective and turbulent transports. The representation of diffusive fluxes in seawater proportional to gradients of salini- ty is, therefore, misleading and, except for the smallest scales, incorrect because the thermodynamical equilibrium state is not incorporated into the equations and cannot be in terms of salinity alone. The different diffusion rates can become impor- tant if the diffusion time scales are comparable or exceed the advective scales.

EQUATION OF STATE FOR SEAWATER

Natural variability in the composition of seawater results in fluctuations in the relationship between density and conduc- tivity that exceed the resolution of measurement. The density of a sample of natural seawater cannot be specified by salinity alone to better than about 50 x 10 -3 kg/m 3 unless correc- tions for composition are made. Brewer and Bradshaw [1975] estimated that the effects of increased alkalinity, total carbon dioxide, and silica content on the conductivity-density re- lationship lead to an underestimate of density from conduc- tivity salinity alone by about 12 x 10 -3 kg/m 3 for the deep North Pacific waters. Such a correction to density is equiva- lent to the difference of potential density over 500 m vertically. Millero et al. [1978] found differences of 17.6 x 10 -3 kg/m 3 between measured densities and densities calculated from con-

ductivity salinity at depths below 1000 m for the North Paci- fic. Poisson et al. [1981] reported differences of measured and calculated densities for surface waters ranging from -6 x 10 -3 kg/m 3 in the western Indian Ocean to -35 x 10 -3 kg/m 3 in the Red Sea.

A precise equation of state, reproducible to better than 10 x 10-3 kg/m 3, requires not only a precise definition of salini-

ty but also of water itself. Both chemical and isotopic vari- ations in natural waters and seawaters affect density at this level of precision. JPOTS had first to define a salinity scale and to choose a reference pure water of specified isotopic composition before considering candidate formulas for the density and specific volume. Procedures for defining the prac- tical salinity scale and the equation of state were set out in the eighth report [UNESCO, 1978] of the panel. Five data sets were identified as the basis for developing the equation of state, using the general form adopted by Chen and Millero [1976], similar to the earlier equation used by Ekman [1908]. Ekman incorporated the effects of pressure in a mean com- pression coefficient, whereas Chen and Millero fitted the mean

FOFONOFF: PRACTICE OF OCEANOGRAPHY 3337

TABLE 3. Coefficients for the International Equation of State for Seawater EOS 80

A B C D

t o +999.842594' +8.24493E-1 t • + 6.793952E-2 -4.0899 E-3 t 2 --9.095290E-3 + 7.6438 E-5 t 3 4-1.001685E-4 - 8.2467 E-7 t 4 - 1.120083E-6 + 5.3875 E-9 t 5 + 6.536332E-9

- 5.72466E-3

+ 1.0227 E-4 - 1.6546 E-6

+4.8314E-4

E F G

t o 19652.21' +54.6746 t • + 148.4206 -0.603459 t 2 -2.327105 + 1.09987E-2 t 3 + 1.360477E-2 -6.1670 E-5 t 4 - 5.155288E-5

+ 7.944 E-2 + 1.6483E-2 - 5.3009E-4

H I J

t o + 3.239908* + 2.2838E-3 t • + 1.43713E-3 - 1.0981E-5 t 2 4-1.16092E-4 - 1.6078E-6 t 3 - 5.77905E-7

+ 1.91075E-4

M N

+ 8.50935E-5' -9.9348E-7 -6.12293E-6 + 2.0816E-8

5.2787 E-8 + 9.1697E-10

V(S, t, p)= V(S, t, 0)[1 - p/K(S, t, p)] p(S, t, 0): I/V(S, t, 0) -- ,4 4- BS 4- CS 3/2 4- DS 2

K(S, t, p) = E + FS + GS 3/2 + (H + IS + JS3/2)P + (M + NS)p 2

where the coefficients are polynomials in temperature t. Units: sa- linity, PSS 78; temperature, øC; pressure, bars;density, kg/m3; specif- ic volume, m

*Coefficients modified for calculation of steric anomaly (Table 4).

or secant bulk modulus, the reciprocal of mean compression, to the data.

The data sets consist of measurements made with a mag- netic float densimeter of the density difference between seawa- ter and pure water at atmospheric pressure [Millero et al., 1976] and at elevated pressures [Chen and Millero, 1976]; measurements of compression and thermal expansion by Brad- shaw and $chleicher [1970, 1976]; and compressibility coef- ficients estimated from sound speed measurements of Chen et al. [1977] and Chen and Millero [1978]. A sixth data set was added [UNESCO, 1979; Poisson et al., 1980] to improve the accuracy of the equation at atmospheric pressure at salinity of 35 and temperatures above 25øC. The revised formula was completed by Millero and Poisson [1981] and recommended by JPOTS in the tenth report [UNESCO, 1981a].

Densities of seawater are not determined relative to abso-

lute standards of mass and length but rather as differences from a defined pure water standard of known isotopic compo- sition. Crai•t [1961] described a standard mean ocean water (SMOW) with isotopic composition referred to a National Bureau of Standards reference sample. Menache [Girard and Menache, 1972; UNESCO, 1974] proposed that SMOW be used as the density reference with a provisional value of the maximum density of 999.975 kg/m 3, and the density variation with temperature given by a fifth-order polynomial fitted to Bi•tg's [1967] table of pure water densities. Bigg's table was computed as a weighted mean of measurements made by

Chappuis [1907] and Thiesen et al. [1900] on pure water. Bigg adjusted the earlier values to take into account the re- definition of the liter in 1964. The polynomial fitted to Bigg's values was scaled by a constant factor to adjust the maximum density from 999.972 to 999.975 kg/m 3 for SMOW. Menache [UNESCO, 1976a, Annex 5] estimated that the uncertainty in the maximum density of SMOW did not exceed 3 x 10 -3 kg/m 3. The uncertainty is of little consequence in most ocean- ographic work, provided the same reference is used consis- tently.

Density differences between seawater and SMOW were computed from the data of Millero et al. [1976] and Poisson et al. [1980]. Polynomials in temperature and salinity (half powers) were fitted to the differences to yield a 1-atmosphere equation with a standard error (rms residuals) of 3.6 x 10-3 kg/m 3 [Millero and Poisson, 1981].

The pressure dependence was developed as a mean bulk modulus for pure water K•, based on Kell's [1975] and iso- thermal compressibilities near 1 atmosphere from sound speed measurements of Del Grosso [1970] and Del Grosso and Mader [1972]. The bulk modulus for seawater was determined from sound speed measurements of Millero and Kubinski [1975]. At higher pressures the differences from pure water, V(S, t, p)- V(O, t, p), were calculated from measurements by Chen and Millero [1977] and Bradshaw and Schleicher [1976]. The differences were then fitted by a polynomial in salinity (half powers), temperature, and pressure [Millero et al., 1980]. The overall standard deviation for V(S, t, p) is 9.0 x 10 -9 m3/kg. The final equation of state was adopted by JPOTS in 1981 [UNESCO, 1981a]. Coefficients for the complete equa- tion are given in Table 3.

DENSITY AND STERIC ANOMALIES

For most oceanographic usage the full value of density or specific volume is unnecessary. Because the maximum vari- ation in magnitude over the oceanic range of salinity, temper- ature, and pressure is only 7%, a considerable improvement in numerical resolution is achieved by using anomalies of specific volume and density. The specific volume (steric) anomaly rS(m3/kg) and density anomaly 7(kg/m 3) are defined by

(5(S, t, p)= V(S, t, p)- V(35, O, p) (17)

7(S, t, p)- p(S, t, p)- 1000.0 kg/m 3

where V(35, 0, p) is given by a separate formula (Table 4). Fofonoff and Millard [1984] computed coefficients of the dif- ference formulas needed to retain numerical precision in

TABLE 4. Coefficients for V(35, 0, p) and K(35, 0, p)

Coefficient

V(35, 0, p) = V(35, 0, 0)[1 - p/K(35, O, p)] K(35, 0, p) = E35 + H35 p 4- M35P 2 AK(S, t, p)= K(S, t, p)- K(35, 0, p)

= AE35 4- lkH35P + AM35P 2

E35 = 4- 21582.27 AE35 = - 1930.06 H35 -- 4- 3.359406 AH35 = -0.1194975 M35 -- 4- 5.03217E-5 AM35 = + 3.47718E-5

V(35, 0, 0)= 972.662039 x 10 -6 m3/kg p(35, 0, 0)= 1028.106331 kg/m 3

p(35, 0, 0) - p(0, 0, 0) -- + 28.263737 kg/m 3

Units'pressure, bars.

3338 FOFONOFF: PRACTICE OF OCEANOGRAPHY

TABLE 5. Steric Anomaly and Specific Volume Differences Between Knudsen-Ekman and EOS 80 Equations of State

for Seawater

AV, Ag, x 10 -9 m3/kg x 10 -9 m3/kg

S/t OøC 5øC 10øC 15øC 20øC

p = 0 dbar 34 - 1.6 -2.3 -0.3 0.7 0.1 35 - 18.9 0.0 - 1.0 1.2 2.5 2.0 36 1.4 0.1 2.4 4.0 3.5

p- 2000 dbar 34 - 1.6 -0.3 4.3 6.8 6.4 35 -43.4 0.0 1.2 5.9 8.6 8.2 36 1.4 2.4 7.3 10.1 9.8

p = 4000 dbar 34 - 1.6 -0.9 3.4 5.4 5.1 35 -41.6 0.0 0.7 5.1 7.1 6.9 36 1.5 2.0 6.5 8.7 8.5

/xV = V•E- VEoss0

A6 = C•KE -- 6EO s SO

evaluating 6 and 7. Given the two anomalies, the values of specific volume V and density p are recoverable to higher precision than from the full formula for a typical single- precision computation using a 32-bit computer word length. The increased numerical resolution is particularly important in computation of static stability where small vertical gradi- ents of density must be resolved.

A significant departure from historical usage is taken in the definition of density anomaly 7. Previously, a specific gravity anomaly a t [Knudsen, 1901], defined as

at-- 1000 (p(S, t, O)/p(O, tmax, 0)- 1) (18)

was used, where tma x is the temperature of the density maxi- mum of pure water at atmospheric pressure. The density anomaly 3,, corresponding to at, is

3,(S, t, O)= p(O, t .... 0)(1 + at/1000)- 1000.0 kg/m 3 (19)

Using the accepted value of p(0, t .... 0) = 999.975 kg/m 3 [UNESCO, 1974], (19) becomes

7(S, t, 0)= 0.999975 a,- 0.025 kg/m 3 (20)

The new definition of density anomaly yields values of 3' that are numerically lower (the dimensions are different) than at by

a nearly constant amount. The difference must be taken into account for comparison of properties on potential density sur- faces and similar graphical or numerical displays using a t as a variable. Steric anomalies computed from the Knudsen- Ekman equation of state [Fofonoff, 1962] differ by less than the precision of measurement from EOS 80 values. A compari- son is given in Table 5. Over most of the open ocean range of variation, the differences are 2 x 10 -9 m3/kg or less and affect dynamic height or geopotential anomaly calculations by about 1 dynamic centimeter (0.1 J/kg) or less. As errors of measurement and random variations produced by internal waves are greater, the new salinity scale and equation of state do not significantly affect dynamic computations in agreement with the assessment of Reid [1958].

SPECIFIC HEAT C•,

Specific heat Cp is defined as the heat in Joules required to raise the temperature of 1 kg of seawater IøC at constant pressure.

Millero et al. [1973] measured Cp for standard seawater, diluted with pure water or concentrated by evaporation, over a chlorinity range of 0-229/00 and a temperature range of 5- 35øC. The measurements agreed with earlier determinations to within 2 J/(kgøC), except at low temperatures, where differ- ences from values of Cox and Smith [1959] were as high as 6 J/(kgøC).

The formulas given by Millero et al. [1973] were converted to salinity by using the factor 1.80655 consistent with PSS 78 [UNESCO, 1981c, p. 118] for the 1-atmosphere terms.

No direct measurements of specific heat at elevated pressure are available for seawater. The pressure dependence is com- puted by integration of the thermodynamic relationship

c•2V t•C• - r (21) c•p 63t 2

Io • c• 2 V C•,(S, t, p) = C•(S, t, O) - (t + 273.15) • dp (22) Using V = V0(1- p/K), the derivative can be expressed in terms of V0, K, and derivatives with respect to temperature. As pressure enters only in terms of the form pm/K", where rn and n are integers, the entire integral can be evaluated explicitly in terms of quadratic trinomial integrations of the form

;0 In m-- pm/Kn dp (23)

TABLE 6. Precision of Data for Specific Heat of Seawater

Number of Minimum Interval Maximum Values Total

Standard

Deviation

C(S, t, O)from Millero et al. [1973] Chlorinity 0 variable 22%0 13 52

t 5 10 35øC 4

AC• (from EOS 80) t -2 2 42øC 23

p 50 50 1000 bars 20 460

AC 2 (from EOS 80) S 5 5 40 8 t 0 5 40øC 9

p 100 100 1000 bars 10 720

0.5 J/(kgøC) at 5øC 0.2 J/(kgøC) at 15-35øC

0.074 J/(kgøC)

0.062 J/(kgøC)

Cp(S, t, p) = Cp(S, t, 0) + AC•(0, t, p) + AC2(S , t, p)

FOFONOFF' PRACTICE OF OCEANOGRAPHY 3339

TABLE 7. Specific heat of Seawater Cp(S, t, p) in J/(kgøC)

A B C

4127.4

-3.720283

+0.1412855 - 2.654387E-3

+ 2.093236E-5

- 7.643575 + 0.1770383 + 0.1072763 -4.07718E-3 - 1.38385E-3 - 5.148 E-5

D E F

t o -4.9592 E-1 +4.9247 E-3 t • + 1.45747E-2 - 1.28315E-4 t 2 -- 3.13885E-4 + 9.802 E-7 t 3 +2.0357 E-6 +2.5941 E-8 t 4 + 1.7168 E-8 -2.9179 E-10

- 1.2331E-4

- 1.517 E-6

+ 3.122 E-8

G H I

t o + 2.4931 E-4 -2.9558 E-6 t • - 1.08645E-5 + 1.17054E-7 t 2 + 2.87533E-7 -2.3905 E-9 t 3 -4.0027 E-9 + 1.8448 E-11 t 4 + 2.2956 E-11

+9.971E-8

J K M

t o -5.422 E-8 +5.540 E-10 t • + 2.6380E-9 - 1.7682E-11 t 2 -6.5637E-11 + 3.513 E-13 t 3 +6.136 E-13

- 1.4300E-12

Cp(S, t, p)= A + BS + CS 3/2 + (D + ES + FS3/2)p to

t • + (G + HS + IS3/2)p 2 t2 + (d + KS + MS3/2)p 3 t 3

Coefficients are polynomials in temperature t. Units' salinity, PSS 78' t 4 temperature, øC; pressure, bars. t 5

[Gradsteyn and Ryzhik, 1965]. In computing the pressure de- pendence the expressions were evaluated explicitly in double t o precision to serve as an independent check on direct compu- t•

t 2 tation by finite differencing and numerical integration of (22). t3

Tables of specific heat values were generated from the exact t 4 formulas and polynomial expressions fitted to the tables in t s two steps. The temperature and pressure dependence at $ -- 0 was evaluated by fitting to tables of AC• = C,(0, t, p)- C,(0, t, 0). to

The salinity dependence was obtained by fitting additional terms to a table of t 2

t 3

AC2 = [C,(S, t, p)- C,(S, t, 0)] - It,(0, t, p)- C,(0, t, 0)] t 4 so that

C.(S. t. p)= c.(s. t. 0) + Ac•(0. t. p) + AC2(S. t. p) (24) to

Standard deviations of residuals from the fit are given in Table t 2 6. The polynomial fits for AC• and AC 2 together with the t 3 polynomials for C,(S, t, 0) given by Millero et al. [1973] pro- vide specific heat values over the full range of salinity temper- ature and pressure. The coefficients are given in Table 7. The polynomial terms giving the pressure dependence were fitted with higher precision so that the 1-atmosphere portion can be replaced without having to refit the pressure terms computed from EOS 80. Tables of C, are given by Fofonoff and Millard [1984].

ADIABATIC LAPSE RATE F

The adiabatic lapse rate F(S, t, p), defined as the change of temperature per unit pressure change for an adiabatic dis-

placement, is required for computation of static stability and comparison of water types in the ocean. From thermodynamic considerations the lapse rate is given by

3V

rfs, t, p)= (T/Cp) c3t (25) and can be computed from the equation of state and specific heat Cp.

Bryden [1973] developed a formula for F from Bradshaw and $chleicher's [1970] measurements of thermal expansion of seawater. As these data were a subset of their thermal ex-

pansion data for EOS 80, the agreement with the new equa- tion can be expected to be good. Caldwell and œide [1980] estimated adiabatic lapse rates in seawater by measuring di- rectly the temperature change produced by rapid changes of pressure. The two methods agree within the estimated uncer- tainties. However, the accuracies of the two techniques are not sufficient to permit a critical comparison.

For comparison with the Bryden [1973] and Caldwell and Eide [1980] formulas, tables of F were computed from the thermodynamical equations by using EOS 80 and fitted by standard least squares regression techniques with polynomials in S, t, and p. Sufficient terms were included to reduce the

TABLE 8. Adiabatic Lapse Rate FEo sSo

A B C D

-4.4017 +0.264117 -6.298 E-3 + 1.159313 - 8.4901E-3 + 1.3742E-4 - 1.42752E-2 +9.5714E-5 +2.4153E-6 + 1.94206E-4 - 3.483 E-7 - 1.5540E-8 -4.635 E-7 -7.244 E-9

- 1.2660 E-8

- 2.926 E-5

E F G H

+2.4624 E-2 -2.027 E-4 - 9.5979 E-4 + 9.5596E-6

+ 1.8232 E-5 -2.283 E-7 - 3.0623 E-7 + 4.133 E-9 + 3.5496 E-9 - 5.392 E-11 -2.3005 E-11 +3.717 E-13

+ 4.4614E-6 - 3.7257E-7

+ 8.3866E-9 -9.8194E-11

+2.9171E-7 -4.5887E-9

I J K

-4.9799 E-6 +3.965 E-8 + 2.0840 E-7 -2.251 E-9 -3.5905 E-9 +4.143 E-11 +6.4237 E-11 -4.234 E-13 - 5.9233 E- 13

- 1.9155E-9

+ 6.5229E-11

L M

- 1.1437 E-9

+4.8665 E-11 - 1.1803 E-12

+ 1.3351 E-14

+ 8.0102E-12

N

t o + 3.5773 E-13 t; -9.0147 E-15

F(S, t, p) = A + BS + CS 3/2 + DS 2 + (E + FS + G$ 3/2 + HS2)p + (I + dS + KS3/2)p 2 + (L + MS)p 3 + (N)p •

Units: salinity, PSS 78; temperature, øC; pressure, bars; F, 10-4øC/bar.

3340 FOFONOFF: PRACTICE OF OCEANOGRAPHY

TABLE 9. Comparison of Lapse Rate F at S = 35, t = 0øC

Pressure, FBR -- FEOS 80, FCE -- FEOS 80, bars øC/bar øC/bar

0 7.8 x 10 -6 28.5 x 10 -6 200 4.2 23.5 400 - 0.5 15.7 600 - 5.0 8.6 800 -8.7 4.8

1000 - 11.7 6.2

BR - Bryden [1973]' CE - Caldwell and Eide [1980].

fitting errors to about 0.1 x 10-7øC/dbar. The coefficients are given in Table 8. Bryden estimated his fitting error to be about 3.4 x 10- 7øC/dbar and the uncertainty of the estimate of F to range from 27 to 53 x 10-vøC/dbar or about 2% of F. Cald- well and Eide obtained a fitting error of 5.5 x 10-vøC/dbar corresponding roughly to their estimated uncertainty. A com- parison with I'EOS8 o at S = 35 is made in Table 9. The differ- ences are within the rather conservative limits given by Bryden but large compared with the fitting errors. Bryden's values are closer to FEOS8O, as expected, but show a similar trend with pressure as the Caldwell and Eide residuals.

Because the differences in lapse rates yield adiabatic temper- ature changes that are within the precision of temperature measurements (2-5 x 10-3øC), Fofonoff and Millard [1984] selected the Bryden [1973] formula for computing lapse rates and potential temperatures. The introduction of a new formu- la would yield insignificant improvement in accuracy of poten- tial temperatures and may be in need of further revision in the near future if the differences between the directly measured lapse rates of Caldwell and Eide [1980] and the computed rates are confirmed.

POTENTIAL TEMPERATURE 0

Potential temperature O(S, t, p, p,) is defined to be the tem- perature an element of seawater would have if moved adia- batically and with no change of salinity from an initial pres- sure Po to a reference pressure p, that may be greater or less than Po. This definition is more general than the classical defi- nition given by Helland-Hansen [1912] and requires an ex- plicit choice of the reference pressure p,. However, it simplifies the computation of static stability, Brunt-V/•is/•l/• frequency, and potential density or steric anomaly for comparison of water types.

Under this definition the potential temperature is given by

O(So, to, Po, P,) = to + O(So, to, Po, P), P) dp (26)

where F is the adiabatic lapse rate. Fofonoff [1977] adopted a fourth-order Runge-Kutta inte-

gration scheme to evaluate the implicit integral in (26). He showed that the potential temperature is given to a numerical precision of 0.1 x 10-3øC for the maximum integration step of 10,000 decibars.

Thus the integration can be carried out efficiently in a single step, making it unnecessary to develop separate formulas for potential temperature. A comparison of potential temper- atures computed with the Runge-Kutta algorithm by using Bryden's [1973] and Caldwell and Eide's [1980] formulas is made in Table 9.

Table 10 contains differences from potential temperatures computed from EOS 80 by using the thermodynamical re- lationships. Agreement at 10øC is better for the direct measurements of Caldwell and Eide [1980]. At 0øC, agreement is better with values obtained from Bryden's [1973] formula. As the differences between the Bryden formula and EOS 80 in the oceanic range are comparable to the precision of temper- ature measurements made at sea, no real improvement in ac- curacy is gained by introducing a more precise formula for F. As a consequence, Fofonoffand Millard [1984] recommended using the simpler Bryden formula for F for computation of potential temperature.

The change of salinity scale to PSS 78 produces negligible changes in F or potential temperature 0. The maximum differ- ence in 0 was estimated to be 0.3 x 10-3øC, using Bryden's [1973] F for Po = 10,000 dbar, p, = 0 dbar.

BRUNT-V•IS•L• FREQUENCY

An element of seawater displaced adiabatically and isen- tropically to z from its equilibrium level z 0 in a stratified layer will have a density difference Ap from its surroundings of

Ap = p(So, O(So, to, Po, P), P)- p(S, t, p) (27)

and will be acted upon by a restoring buoyancy force F per unit mass, given by

• = --(Ap/p)• (28)

where g is gravity. The mass element will be subjected to a vertical acceleration 5', where

= + r = (29)

= --N2(z -- Zo) = _N2Az

and, using the hydrostatic relationship Ap = -pgAz,

/2= +(g/p)Ap/Az =-g2Ap/Ap (30)

The Brunt-V•iis/il/i frequency N is a measure of the stratifi- cation and the high-frequency limit for internal waves that can be supported by the vertical potential density gradient.

The dependence of N 2 on thermal expansion and saline contraction coefficients is obtained by expanding (27) in a

TABLE 10. Potential Temperature Differences 0--0EOS80 for S= 35 and Reference Pressure p, = 0 bar

OøC 10øC

Pressure, bars CE, x 10-3øC BR, x 10-3øC CE, x 10-3øC BR, x 10-3øC

200 -5.3 - 1.3 - 1.7 1.0 400 - 9.2 - 1.8 - 2.0 2.0 600 - 11.8 - 1.6 - 1.1 3.1 800 - 13.4 -0.7 +0.1 4.3

1000 - 15.1 + 0.5 + 0.8 5.5

BR - Bryden [1973]' CE - Caldwell and Eide [1980].

FOFONOFF' PRACTICE OF OCEANOGRAPHY 334i

Taylor series about P0 as follows

8p p(S o, O, p) = p(S o, t o, P0) + •-/A0

8p

+ •pp Ap + 0(Ap 2) (31)

8p p(S, t, p)= p(S o, to, Po) + • AS

8p 8p

+ •- At + •pp Ap + 0(Ap 2) so that

Ap

---- •--•pp+• •pp-F +0(Ap) (32)

The compressibility terms cancel exactly. For computational convenience the expression for N 2 is

converted to use specific volume anomaly, i.e.,

N 2 = --g2Ap/Ap - +p2g2AV/Ap = p2g2AS/Ap

= p29218(So, O(So, to, Po, P), P)- 8(S, t, p)]/Ap (33)

The use of potential steric anomaly 8 to compute N provides the maximum numerical resolution and reduces the compu- tation to two evaluations of steric anomaly. Fofonoff and Mil- lard [1984] give algorithms for computing p, l/, 8. The pro- cedure can be elaborated by referencing a series of observa- tions over a suitable pressure interval to a single pressure and fitting a least squares curve to the potential steric anomalies. Such a method [Bray and Fofonoff, 1981] has the advantage of vertical averaging to improve the estimate of N 2.

DISCUSSION

The need to establish standards for measurement of salinity with a precision matching the sensitivity of present-day con- ductivity bridges has forced abandonment of scales based on natural seawaters in favor of a more rigidly and precisely defined practical scale referenced to a conductivity standard. The practical scale will continue to be used until measurement techniques advance in precision and resolution beyond the present standards. As salinity is combined with temperature and pressure to compute density in many applications, the relationship between conductivity and density will need to be explored further to establish the range of compositional vari- ations in the world oceans and to establish base lines for

monitoring time-dependent changes in the relationship.

Acknowledgment. This article is made possible by the generous support of the Office of Naval Research under contract N00014-76-C- 0197, NR 083-400, and by the interest and motivation provided by members of JPOTS and SCOR Working Group 51 on evaluation of CTD data. The invitation to contribute to a volume dedicated to R.

O. Reid provided an opportunity to review some of the historical development of the new salinity scale and equation of state of seawa- ter.

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(Received October 16, 1984; accepted October 27, 1984.)